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An object in a free coproduct completion is Galois closed if it has no nontrivial coverings, i.e every covering morphism is split by the identity.

An object of a Galois category is a Galois object if its quotient by its automorphism group is the terminal object.

What is the connection between Galois objects and Galois closed objects?

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  • $\begingroup$ One might ask here whether every Galois closed object is a Galois object. Do you mean anything else? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 15, 2016 at 19:25
  • $\begingroup$ @მამუკაჯიბლაძე that and any other implications between them under reasonable conditions. $\endgroup$
    – Arrow
    Commented Jul 16, 2016 at 6:54
  • $\begingroup$ I see only one other implication, and it is false under any reasonable conditions. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 16, 2016 at 7:18
  • $\begingroup$ @მამუკაჯიბლაძე Could you explain/give intuition as to why the other implication is false? What about the implication in your first comment? $\endgroup$
    – Arrow
    Commented Jul 16, 2016 at 7:44
  • $\begingroup$ Consider $G$-sets (a typical example): a Galois object is one with all of its orbits isomorphic to each other, i. e. all stabilizers conjugate to each other. This is a Galois closed object iff all these stabilizers are trivial. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jul 16, 2016 at 9:09

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